Illumination of complex 1-unconditional convex bodies

אלון שכטר

Illumination of complex 1-unconditional convex bodies

Primarily AI-generated textHuman understanding: some partsmath.MG — Metric Geometry

Contributed by אלון שכטר ↗

Version 1 / Nov 02, 2026 / CC BY 4.0

Abstract

We study ordinary and fractional illumination of complex

1

-unconditional convex bodies. In every complex dimension

n

we give an explicit ordinary bound asymptotic to

3^{n+1}/\sqrt{\pi n}

, and prove $\ill^*(K)\le b_*^n$ and $\ill(K)=O(n\log(n+1)b_*^n)$, where

b_*=2.4805534697\ldots

is defined by an explicit scalar equation. For normalized non-polydiscs in dimensions two, three and four, the sharp ordinary bounds are

6

,

14

and

30

. More generally, a full coordinate

(n-1)

-polydisc gives the sharp bound

2^{n+1}-2

in every dimension

n\ge2

. In dimension two we prove the fractional bound

4

and classify equality by the two maximal supporting coefficients at the coordinate axes. If every extreme point has all coordinates nonzero, we obtain the fractional bound

2^n

, with equality exactly for the polydisc, and a quantitative strict bound otherwise. Appendices give the exact sizes of simultaneous illuminating families in dimensions one through five, the exact finite verification used in dimension four, and a numerical comparison of the general estimates through dimension

100

.

Provenance statement

Results generated with Chatgpt Astra-6. most of the content was examined by a human.

References

S. Artstein-Avidan and B. A. Slomka, On weighted covering numbers and the Levi-Hadwiger conjecture , Israel J. Math. 209 (2015), no. 1, 125--155. doi:10.1007/s11856-015-1213-5 (https://doi.org/10.1007/s11856-015-1213-5). For the locators used here, see arXiv:1310.7892v2 (https://arxiv.org/abs/1310.7892v2).arXiv DOI
V. Boltyanski and H. Martini, Illumination of direct vector sums of convex bodies , Studia Sci. Math. Hungar. 44 (2007), no. 3, 367--376. doi:10.1556/SScMath.44.2007.3.6 (https://doi.org/10.1556/SScMath.44.2007.3.6).DOI
P. Hall, On representatives of subsets , J. London Math. Soc. s1-10 (1935), no. 1, 26--30. doi:10.1112/jlms/s1-10.37.26 (https://doi.org/10.1112/jlms/s1-10.37.26).DOI
M. Naszódi, Fractional illumination of convex bodies , Contrib. Discrete Math. 4 (2009), no. 2, 83--88. doi:10.55016/ojs/cdm.v4i2.62022 (https://doi.org/10.55016/ojs/cdm.v4i2.62022).DOI
L. Rotem, A. Schejter, and B. A. Slomka, The complex Illumination problem , Combinatorica 46 , article 3 (2026). doi:10.1007/s00493-025-00195-7 (https://doi.org/10.1007/s00493-025-00195-7). The theorem, proposition and remark numbers cited here refer to the published article.DOI
L. Rotem, A. Schejter and B. Slomka, Illumination of Real and Complex 1-Symmetric Convex Bodies , unpublished manuscript, October 2026, 22 pages. The locators used here refer to the version with Theorems 1.5--1.8, Lemma 4.3 and Proposition 4.4.
W. R. Sun and B.-H. Vritsiou, Illuminating $1$ -unconditional convex bodies in $\mathbb R^3$ and $\mathbb R^4$ , and certain cases in higher dimensions , Canad. J. Math., accepted manuscript posted online 4 August 2025. doi:10.4153/S0008414X25101260 (https://doi.org/10.4153/S0008414X25101260). The theorem numbers used here refer to arXiv:2407.11331v1 (https://arxiv.org/abs/2407.11331v1) (2024).arXiv DOI
W. R. Sun and B.-H. Vritsiou, On the illumination of $1$ -symmetric convex bodies , arXiv:2407.10314v1 (https://arxiv.org/abs/2407.10314v1) (2024).arXiv

Version history

v1 · 34 pagesSubmitted by אלון שכטרInitial depositCurrentNov 01, 2026